Periodic functions

Where \(k, N\in\mathbb{z}\) and \(T\in\mathbb{R}\)

Sum of Two Periodic Signals

If \(\left( x_1(t) \right)\) and \(\left( x_2(t) \right)\) are periodic with periods \(\left( T_1 \right)\) and \(\left( T_2 \right)\):

\[ x_1(t + T_1) = x_1(t), \quad x_2(t + T_2) = x_2(t) \]

The sum of both signals is:

\[ x(t) = x_1(t) + x_2(t) \]

Condition for the Periodicity of the Sum

For \(\left( x(t) \right)\) to be periodic, there must exist a common period \(\left( T \right)\) such that:

\[ T = k_1 T_1 = k_2 T_2 \]

where \(\left( k_1, k_2 \right)\) are positive integers.

Common Period and Least Common Multiple

The smallest common period is the least common multiple (lcm) of \(\left( T_1 \right)\) and \(\left( T_2 \right)\):

\[ T = \operatorname{lcm}(T_1, T_2) \]

If the ratio of the periods is a rational number:

\[ \frac{T_1}{T_2} \in \mathbb{Q} \]

Then, the sum \(\left( x_1(t) + x_2(t) \right)\) will be periodic.

If the ratio is irrational, the resulting signal will not be periodic.

Example